Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8
January 24, 2007
Textbook §2.6
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 1
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Lecture Overview
1
Recap
2
Branch & Bound
3
A∗ Tricks
4
Other Pruning
5
Backwards Search
6
Dynamic Programming
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 2
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Lecture Overview
1
Recap
2
Branch & Bound
3
A∗ Tricks
4
Other Pruning
5
Backwards Search
6
Dynamic Programming
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 3
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
A∗ Search Algorithm
A∗ is a mix of lowest-cost-first and Best-First search.
It treats the frontier as a priority queue ordered by f (p).
It always selects the node on the frontier with the lowest
estimated total distance.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 4
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Analysis of A∗
Let’s assume that arc costs are strictly positive.
Completeness: yes.
Time complexity: O(bm )
the heuristic could be completely uninformative and the edge
costs could all be the same, meaning that A∗ does the same
thing as BFS
Space complexity: O(bm )
like BFS, A∗ maintains a frontier which grows with the size of
the tree
Optimality: yes.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 5
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Optimal Efficiency of A∗
In fact, we can prove something even stronger about A∗ : in a
sense (given the particular heuristic that is available) no
search algorithm could do better!
Optimal Efficiency: Among all optimal algorithms that start
from the same start node and use the same heuristic h, A∗
expands the minimal number of nodes.
problem: A∗ could be unlucky about how it breaks ties.
So let’s define optimal efficiency as expanding the minimal
number of nodes n for which f (n) 6= f ∗ , where f ∗ is the cost
of the shortest path.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 6
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Why is A∗ optimally efficient?
Theorem
A∗ is optimally efficient.
Let f ∗ be the cost of the shortest path to a goal. Consider any
algorithm A0 which has the same start node as A∗ , uses the same
heuristic and fails to expand some node n0 expanded by A∗ for which
cost(n0 ) + h(n0 ) < f ∗ . Assume that A’ is optimal.
Consider a different search problem which is identical to the original
and on which h returns the same estimate for each node, except that
n0 has a child node n00 which is a goal node, and the true cost of the
path to n00 is f (n0 ).
that is, the edge from n0 to n00 has a cost of h(n0 ): the heuristic is
exactly right about the cost of getting from n0 to a goal.
A0 would behave identically on this new problem.
The only difference between the new problem and the original problem
is beyond node n0 , which A0 does not expand.
Cost of the path to n00 is lower than cost of the path found by A0 .
This violates our assumption that A0 is optimal.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 7
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Lecture Overview
1
Recap
2
Branch & Bound
3
A∗ Tricks
4
Other Pruning
5
Backwards Search
6
Dynamic Programming
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 8
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Branch-and-Bound Search
A search strategy often not covered in AI, but widely used in
practice
Uses a heuristic function: like A∗ , can avoid expanding some
unnecessary nodes
Depth-first: modest memory demands
in fact, some people see “branch and bound” as a broad family
that includes A∗
these people would use the term “depth-first branch and
bound”
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 9
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Branch-and-Bound Search Algorithm
Follow exactly the same search path as depth-first search
treat the frontier as a stack: expand the most-recently added
node first
the order in which neighbors are expanded can be governed by
some arbitrary node-ordering heuristic
Keep track of a lower bound and upper bound on solution
cost at each node
lower bound: LB(n) = cost(n) + h(n)
upper bound: U B = cost(n0 ), where n0 is the best solution
found so far.
if no solution has been found yet, set the upper bound to ∞.
When a node n is selected for expansion:
if LB(n) ≥ U B, remove n from frontier without expanding it
this is called “pruning the search tree” (really!)
else expand n, adding all of its neighbours to the frontier
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 10
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Branch-and-Bound Analysis
Completeness: no, for the same reasons that DFS isn’t
complete
however, for many problems of interest there are no infinite
paths and no cycles
hence, for many problems B&B is complete
Time complexity: O(bm )
Space complexity: O(bm)
Branch & Bound has the same space complexity as DFS
this is a big improvement over A∗ !
Optimality: yes.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 11
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Lecture Overview
1
Recap
2
Branch & Bound
3
A∗ Tricks
4
Other Pruning
5
Backwards Search
6
Dynamic Programming
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 12
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Other A∗ Enhancements
The main problem with A∗ is that it uses exponential space.
Branch and bound was one way around this problem. Are there
others?
Iterative deepening
Memory-bounded A∗
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 13
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Iterative Deepening
B & B can still get stuck in cycles
Search depth-first, but to a fixed depth
if you don’t find a solution, increase the depth tolerance and
try again
of course, depth is measured in f value
Counter-intuitively, the asymptotic complexity is not changed,
even though we visit nodes multiple times
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 14
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Memory-bounded A∗
Iterative deepening and B & B use a tiny amount of memory
what if we’ve got more memory to use?
keep as much of the fringe in memory as we can
if we have to delete something:
delete the oldest paths
“back them up” to a common ancestor
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 15
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Lecture Overview
1
Recap
2
Branch & Bound
3
A∗ Tricks
4
Other Pruning
5
Backwards Search
6
Dynamic Programming
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 16
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Non-heuristic pruning
What can we prune besides nodes that are ruled out by our
heuristic?
Cycles
Multiple paths to the same node
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 17
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Cycle Checking
s
You can prune a path that ends in a node already on the
path. This pruning cannot remove an optimal solution.
Using depth-first methods, with the graph explicitly stored,
this can be done in constant time.
For other methods, the cost is linear in path length.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 18
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Multiple-Path Pruning
s
You can prune a path to node n that you have already found
a path to.
Multiple-path pruning subsumes a cycle check.
This entails storing all nodes you have found paths to.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 19
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Multiple-Path Pruning & Optimal Solutions
Problem: what if a subsequent path to n is shorter than the first
path to n?
You can remove all paths from the frontier that use the longer
path.
You can change the initial segment of the paths on the
frontier to use the shorter path.
You can ensure this doesn’t happen. You make sure that the
shortest path to a node is found first.
Heuristic function h satisfies the monotone restriction if
|h(m) − h(n)| ≤ d(m, n) for every arc hm, ni.
If h satisfies the monotone restriction, A∗ with multiple path
pruning always finds the shortest path to every node
otherwise, we have this guarantee only for goals
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 20
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Lecture Overview
1
Recap
2
Branch & Bound
3
A∗ Tricks
4
Other Pruning
5
Backwards Search
6
Dynamic Programming
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 21
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Direction of Search
The definition of searching is symmetric: find path from start
nodes to goal node or from goal node to start nodes.
Of course, this presumes an explicit goal node, not a goal test.
Also, when the graph is dynamically constructed, it can
sometimes be impossible to construct the backwards graph
Forward branching factor: number of arcs out of a node.
Backward branching factor: number of arcs into a node.
Search complexity is bn . Should use forward search if forward
branching factor is less than backward branching factor, and
vice versa.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 22
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Bidirectional Search
You can search backward from the goal and forward from the
start simultaneously.
This wins as 2bk/2 bk . This can result in an exponential
saving in time and space.
The main problem is making sure the frontiers meet.
This is often used with one breadth-first method that builds a
set of locations that can lead to the goal. In the other
direction another method can be used to find a path to these
interesting locations.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 23
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Lecture Overview
1
Recap
2
Branch & Bound
3
A∗ Tricks
4
Other Pruning
5
Backwards Search
6
Dynamic Programming
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 24
Recap
Branch & Bound
A∗ Tricks
Other Pruning
Backwards Search
Dynamic Programming
Dynamic Programming
Idea: for statically stored graphs, build a table of dist(n) the
actual distance of the shortest path from node n to a goal.
This can be built backwards from the goal:
0
if is goal(n),
dist(n) =
minhn,mi∈A (|hn, mi| + dist(m)) otherwise.
This can be used locally to determine what to do.
There are two main problems:
You need enough space to store the graph.
The dist function needs to be recomputed for each goal.
Complexity: polynomial in the size of the graph.
Search: Advanced Topics and Conclusion
CPSC 322 Lecture 8, Slide 25